Finite Field Representations by Using Maple

Kahtan H. Alzubaidy

Abstract

Certain Maple procedures are created to represent finite fields in terms of polynomials, cyclic elements , and matrices.

Theoritical Background

A finite field is a field that has a finite number of elements.

Assume that E is a field extension of F. E is a vector space over F, the vectors are the elements of E, the scalars are the elements of F, the product of a scalar s\342\210\210F and a vector v\342\210\210E is sv\342\210\210E.The dimension of this vector space is called the degree of E over F.

Theorem 1[1]

Any finite field is an extension of the field 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

Theorem 2

i) If F is a finite field, then |F|=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 where p is prime and n is a positive integer.

ii) If p is a prime and n a positive integer, then there is a field of order 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

iii)Any two finite fields of the same order are isomorphic.

A finite field of order LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EicEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSVtcm93R0YkNictRiw2J1EibkYnRi9GMkY0RjdGLy8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJy9GNUY5RjcvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYn is denoted by GF(LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EicEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSVtcm93R0YkNictRiw2J1EibkYnRi9GMkY0RjdGLy8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJy9GNUY5RjcvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYn ).

Representations of Finite Fields

I) Polynomials

Theorem 3

Let p be a prime number and n a positive integer. A finite field of order LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EicEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSVtcm93R0YkNictRiw2J1EibkYnRi9GMkY0RjdGLy8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJy9GNUY5RjcvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYn is given as follows: GF(LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EicEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSVtcm93R0YkNictRiw2J1EibkYnRi9GMkY0RjdGLy8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJy9GNUY5RjcvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYn ) =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\342\201\204 \342\214\251m(x)\342\214\252 , where m(x) is an irreducible polynomial of degree n in 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

Elements of GF(LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EicEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSVtcm93R0YkNictRiw2J1EibkYnRi9GMkY0RjdGLy8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJy9GNUY5RjcvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYn ) are of the form 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 +...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 + I , where I=\342\214\251m(x)\342\214\252 is the maximal ideal generated by m(x).

The general element can be written briefly as follows:

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 +...LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUYsNi9GLi8lJWJvbGRHUSV0cnVlRicvRjBRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRkxGMkY1RjdGOUY7Rj1GP0ZBRkQtRiw2L1EiK0YnRkhGS0ZNRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjIyMjIyMmVtRicvRkVGU0ZGRkgvJTBmb250X3N0eWxlX25hbWVHUSVUZXh0RidGS0ZNLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiYUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2LC1GLzYnUSJuRidGMkY1RjdGOi1JI21vR0YkNi9RKCZtaW51cztGJ0YyL0Y4RjxGOi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZYLUkjbW5HRiQ2JlEiMUYnRjJGRkY6RjJGNS8lK2ZvcmVncm91bmRHUSxbMjAwLDAsMjAwXUYnLyUscGxhY2Vob2xkZXJHRjQvJTBmb250X3N0eWxlX25hbWVHUSVUZXh0RidGN0Y6LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGMkZeb0ZGRjo=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1cEdGJDYlLUkjbWlHRiQ2J1EieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2LC1GLzYnUSJuRidGMkY1RjdGOi1JI21vR0YkNi9RKCZtaW51cztGJ0YyL0Y4RjxGOi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZYLUkjbW5HRiQ2JlEiMUYnRjJGRkY6RjJGNS8lK2ZvcmVncm91bmRHUSxbMjAwLDAsMjAwXUYnLyUscGxhY2Vob2xkZXJHRjQvJTBmb250X3N0eWxlX25hbWVHUSVUZXh0RidGN0Y6LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YyRl5vRkZGOg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY6LUkjbW9HRiQ2L1EifkYnLyUlYm9sZEdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZIRitGK0YrLUkjbWlHRiQ2J1EnbW9kdWxvRidGLy8lJ2l0YWxpY0dGOUYyRjVGK0YrLUZMNidRInBGJ0YvRk9GMkY1RitGKy1GLDYvUSRhbmRGJ0YvRjJGNUY3RjpGPEY+RkBGQkZERkZGSUYrRktGKy1GTDYnUSJtRidGL0ZPRjJGNS1JKG1mZW5jZWRHRiQ2Ji1GIzYnLUZMNidRInhGJ0YvRk9GMkY1Ri8vJTBmb250X3N0eWxlX25hbWVHUSVUZXh0RidGMkY1Ri9GMkY1RistRiw2L1EiLkYnRi9GMkY1RjdGOkY8Rj5GQEZCRkRGRkZJRitGK0YvRlxvRjJGNQ==Clearly addition , multiplication and quotient of such elements are carried out by modulo p and modulo m(x).

II) Cyclic Groups

Theorem 4

Let 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GF(LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EicEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSVtcm93R0YkNictRiw2J1EibkYnRi9GMkY0RjdGLy8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJy9GNUY5RjcvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYn ) - {0}. 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

The finite field of order LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZFLUkjbW9HRiQ2L1EifkYnLyUlYm9sZEdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZILUklbXN1cEdGJDYlLUkjbWlHRiQ2J1EicEYnRi8vJSdpdGFsaWNHRjlGMkY1LUYjNictRk82J1EibkYnRi9GUkYyRjVGLy8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRistRk82J1EoaW5kdWNlZEYnRi9GUkYyRjVGK0YrLUYsNi9RI2J5RidGL0YyRjVGN0Y6RjxGPkZARkJGREZGRklGKy1GTzYnUSR0aGVGJ0YvRlJGMkY1RistRk82J1EsaXJyZWR1Y2libGVGJ0YvRlJGMkY1RistRk82J1ErcG9seW5vbWlhbEYnRi9GUkYyRjVGK0YrRistRk82J1EibUYnRi9GUkYyRjUtSShtZmVuY2VkR0YkNiYtRiM2Jy1GTzYnUSJ4RidGL0ZSRjJGNUYvRllGMkY1Ri9GMkY1RistRk82J1EjaXNGJ0YvRlJGMkY1RistRk82J1EoZGVub3RlZEYnRi9GUkYyRjVGK0YrRlxvRitGKy1GTzYnUSNHRkYnRi9GUkYyRjUtRlxwNiYtRiM2LUYrRitGSy1GLDYvUSIsRidGL0YyRjVGNy9GO0YxRjxGPkZARkJGREZGL0ZKUSwwLjMzMzMzMzNlbUYnRitGaG9GW3BGL0ZZRjJGNUYvRjJGNS1GLDYvUSIuRidGL0YyRjVGN0Y6RjxGPkZARkJGREZGRklGK0YvRllGMkY1

An irreducible polynomial m(x) is primitive if x is a primitive element in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2J1EjR0ZGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y3LUkobWZlbmNlZEdGJDYmLUYjNiotSSNtb0dGJDYvUSJ+RidGL0Y1RjgvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTRj8tSSVtc3VwR0YkNiUtRiw2J1EicEYnRi9GMkY1RjgtRiM2JC1GLDYnUSJuRidGL0YyRjVGOC9GNlEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUZANi9RIixGJ0YvRjVGOEZDL0ZGRjFGR0ZJRktGTUZPRlEvRlVRLDAuMzMzMzMzM2VtRidGPy1GLDYnUSJtRidGL0YyRjVGOC1GOzYmLUYjNiQtRiw2J1EieEYnRi9GMkY1RjhGW29GL0Y1RjhGW29GL0Y1RjgtRkA2LVEiLkYnRltvRkNGRUZHRklGS0ZNRk9GUUZURltv

Theorem 5

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

III) Matrices

An associative algebra over a field F is a ring R such that:

i) R is a vector space over F with respect to addition and scalar multiplication by elements of F,

ii) s(uv)=(su)v=u(sv) for all u,v\342\210\210R and s\342\210\210F.

GF(LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EicEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSVtcm93R0YkNictRiw2J1EibkYnRi9GMkY0RjdGLy8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJy9GNUY5RjcvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYn ) is an associative algebra over the field LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2KFErJmludGVnZXJzO0YnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRjEvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjktSSVtcm93R0YkNictRiw2J1EicEYnRi8vRjNGMS9GOFEsYm9sZC1pdGFsaWNGJ0Y6Ri8vJTBmb250X3N0eWxlX25hbWVHUSVUZXh0RidGN0Y6LyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= . Its basis is { 1,x,...,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the set of all n-square matrices over the field F is also an associative algebra .

Theorem 6[2]

Any associative algebra over a field F of dimensional n is isomorphic to a subalgebra of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2J1EiTUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNy8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2Jy1GLDYnUSJuRidGNUY4RjpGPUY1LyUwZm9udF9zdHlsZV9uYW1lR1ElVGV4dEYnL0Y7Rj9GPS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYmLUYjNictRiw2J1EiRkYnRjVGOEY6Rj1GNUZFRkhGPUY1RkhGPUY1RkVGSEY9.

To find the corresponding matrix of a polynomial f(x) in 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

Take the polynomials f(x), xf(x), ...,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

Computations

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJStleGVjdXRhYmxlR0Y9Rjk=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRMFBvbHlub21pYWxUb29sc0YnRi9GMi9GM1Enbm9ybWFsRidGPS1JI21vR0YkNi1RIjpGJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlQvJStleGVjdXRhYmxlR0ZFRj0=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJ0YrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYyLUkjbW9HRiQ2L1EifkYnLyUlYm9sZEdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZIRitGK0YrRistRiw2MEYuLyUlc2l6ZUdRIzE0RidGL0YyRjVGN0Y6RjxGPkZARkJGREZGRkktSSNtaUdGJDYoUSJJRidGTUYvLyUnaXRhbGljR0Y5RjJGNS1GLDYwUSIpRidGTUYvRjJGNS9GOEYxRjovRj1GMUY+RkBGQkZEL0ZHUSwwLjE2NjY2NjdlbUYnL0ZKRmZuRktGSy1GUTYoUStQb2x5bm9taWFsRidGTUYvRlRGMkY1RkstRlE2KFEwUmVwcmVzZW50YXRpb25zRidGTUYvRlRGMkY1LyUrZXhlY3V0YWJsZUdGMS8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RicvRjNRJ25vcm1hbEYn

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRitGKy1GLDYvRi4vJSVib2xkR1EldHJ1ZUYnL0YwUSVib2xkRicvJStmb250d2VpZ2h0R0ZMRjJGNUY3RjlGO0Y9Rj9GQUZERkZGRi8lK2V4ZWN1dGFibGVHRkovJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRi8=

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cp2:=proc(K,L)

[seq(seq([x,y],y in L),x in K)];

end proc;

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IiIq

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cp3:=proc(K,L,M)

local G;

G:=cp2(cp2(K, L), M);

[seq(Flatten(J), `in`(J, G))];

end proc;

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cp4:=proc(K,L,M,N)

local G;

G:=cp2(cp3(K, L, M),N);

[seq(Flatten(J), `in`(J, G))];

end proc;

Zio2JkkiS0c2IkkiTEdGJUkiTUdGJUkiTkdGJTYjSSJHR0YlRiVGJUMkPkYqLUkkY3AyR0YlNiQtSSRjcDNHRiU2JUYkRiZGJ0YoNyMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtX0kqTGlzdFRvb2xzRzYkRjZJKF9zeXNsaWJHRiVJKEZsYXR0ZW5HRiU2I0kiSkdGJS1JI2luR0Y2NiRGP0YqRiVGJUYl

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IiM7

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cp5:=proc(K,L,M,N,O)

local G;

G:=cp2(cp4(K, L,M,N), O);

[seq(Flatten(J), `in`(J, G))];

end proc;

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

Algebraic operations can be performed by using maple command modpol .

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

LCYiIiJGI0kieEc2IkYj

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnbW9kcG9sRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNi4tSSVtc3VwR0YkNiUtRjY2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIrRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZMLyUpc3RyZXRjaHlHRkwvJSpzeW1tZXRyaWNHRkwvJShsYXJnZW9wR0ZMLyUubW92YWJsZWxpbWl0c0dGTC8lJ2FjY2VudEdGTC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRmVuLUkjbW5HRiQ2JFEiMUYnRkhGSEZILUYjNiYtRkU2LVEqJnVtaW51czA7RidGSEZKRk1GT0ZRRlNGVUZXRllGZm5GaG5GL0YyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GRTYtUSIsRidGSEZKL0ZORjFGT0ZRRlNGVUZXL0ZaUSYwLjBlbUYnL0ZnblEsMC4zMzMzMzMzZW1GJy1GOzYlRkEtRiM2JS1GaW42JFEiM0YnRkhGL0YyRmFvRkRGQUZERmhuRmRvRkFGZG8tRmluNiRRIjJGJ0ZIRkhGSC1GRTYtUSI7RidGSEZKRmdvRk9GUUZTRlVGV0Zoby9GZ25RLDAuMjc3Nzc3OGVtRicvJStleGVjdXRhYmxlR0ZMRkg=

LCZJInhHNiIiIiIqJClGIyIiI0YlRiU=

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

LCgiIiJGI0kieEc2IkYjKiQpRiQiIiNGI0Yj

Inverse element

inv:=proc(f,g,x,p);

modpol(1/f ,g,x,p);

end proc;

Zio2JkkiZkc2IkkiZ0dGJUkieEdGJUkicEdGJUYlRiVGJS1JJ21vZHBvbEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYmKiRGJCEiIkYmRidGKEYlRiVGJQ==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LCYiIiJGI0kieEc2IkYj

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

Cyclic representation of G(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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

Cyclic elements

G:=proc(p,n);

[0,[seq(x^r, r = 0 .. p^n-2)][]];

end proc;

Zio2JEkicEc2IkkibkdGJUYlRiVGJTckIiIhJjcjLUkkc2VxRyUqcHJvdGVjdGVkRzYkKUkieEdGJUkickdGJS9GMTtGKCwmKUYkRiYiIiIiIiMhIiJGJUYlRiVGJQ==

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IiIj

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Corresponding field elements

F:=proc(p,n,q,x);

[seq(modpol(G(p,n)[t], q, x, p), t = 1 .. p^n)];

end proc;

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su:=proc(A,B)local t;

t := Search(modpol(A+B, q, x, p), F(p,n,q,x));

if t=1 then 0 else x^(t-2) end if;

end proc;

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KiQpSSJ4RzYiIiIkIiIi

Product of cyclic elements

pr:=proc(A,B)local m;

m := Search(modpol(A*B, q, x, p), F(p,n,q,x));

if m=1 then 0 else x^(m-2) end if;

end proc;

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KiQpSSJ4RzYiIiIjIiIi

CP:=proc(A,B)

local M,P,CP;

M := Matrix(nops(A),nops(B), proc (i, j) options operator, arrow; [A[i], B[j]] end proc);

[seq(seq(M(i, j), i = 1 .. nops(A)), j = 1 .. nops(B))];

end proc;

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Sum table

ST:=proc(C,R,f)

local m,n,M,RR,CC,U;

n:=nops(R);m:=nops(C);M := Matrix(m,n, (x,y)->f(C[x],R[y]));

RR := Matrix(1, n, [R]);

CC:=Matrix(m,1,[seq([x],x=C)]);

U := [sum];

blockmatrix(2, 2, [U, RR, CC, M, U, CC, RR, M]);

end proc;

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PT:=proc(C,R,f)

local m,n,M,RR,CC,U;

n:=nops(R);m:=nops(C);M := Matrix(m,n, (x,y)->f(C[x],R[y]));

RR := Matrix(1, n, [R]);

CC:=Matrix(m,1,[seq([x],x=C)]);

U := [pro];

blockmatrix(2, 2, [U, RR, CC, M, U, CC, RR, M]);

end proc;

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Inverse Table

IT:=proc(n,a)

local L,H;

L:=[seq(a^i, i =0..n-1)];

H:=algsubs(a^(n)=1,[seq(a^(n-i),i=0..n-1)]);

Matrix(2,n,[L,H]);

end proc;

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III) Matrix Representations

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st:=proc(f,q);

sort(rem(f,q,x),x,ascending);end proc;

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mr22:=proc(a,b,x,q) local KK1,KK2;

KK1 := [coeffs(st(a+b*x, q), x)];

KK2 := [coeffs(st(x*(a+b*x), q), x)];

LinearAlgebra[Transpose](Matrix([KK1, KK2]))mod 2;

end proc;

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mr23:=proc(a,b,c,x,q) local KK1,KK2,KK3;

KK1:=[coeffs(st(a+b*x+c*x^2, q), x)];

KK2 := [coeffs(st(x*(a+b*x+c*x^2), q), x)];

KK3 := [coeffs(st(x^2*(a+b*x+c*x^2), q), x)];

LinearAlgebra[Transpose](Matrix([KK1, KK2,KK3]))mod 2;

end proc;

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mr24:=proc(a,b,c,d,x,q) local KK1,KK2,KK3,KK4;

KK1:=[coeffs(st(a+b*x+c*x^2+d*x^3, q), x)];

KK2 := [coeffs(st(x*(a+b*x+c*x^2+d*x^3), q), x)];

KK3 := [coeffs(st(x^2*(a+b*x+c*x^2+d*x^3), q), x)];KK4 := [coeffs(st(x^3*(a+b*x+c*x^2+d*x^3), q), x)];

LinearAlgebra[Transpose](Matrix([KK1, KK2,KK3,KK4]))mod 2;

end proc;

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mr25:=proc(a,b,c,d,e,x,q) local KK1,KK2,KK3,KK4,KK5;

KK1:=[coeffs(st(a+b*x+c*x^2+d*x^3+e*x^4, q), x)];

KK2 := [coeffs(st(x*(a+b*x+c*x^2+d*x^3+e*x^4), q), x)];

KK3 := [coeffs(st(x^2*(a+b*x+c*x^2+d*x^3+e*x^4), q), x)];KK4 := [coeffs(st(x^3*(a+b*x+c*x^2+d*x^3+e*x^4), q), x)];KK5 := [coeffs(st(x^4*(a+b*x+c*x^2+d*x^3+e*x^4), q), x)];

LinearAlgebra[Transpose](Matrix([KK1, KK2,KK3,KK4,KK5]))mod 2;

end proc;

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mr32:=proc(a,b,x,q) local KK1,KK2;

KK1 := [coeffs(st(a+b*x, q), x)];

KK2 := [coeffs(st(x*(a+b*x), q), x)];

LinearAlgebra[Transpose](Matrix([KK1, KK2]))mod 3;

end proc;

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mr33:=proc(a,b,c,x,q) local KK1,KK2,KK3;

KK1:=[coeffs(st(a+b*x+c*x^2, q), x)];

KK2 := [coeffs(st(x*(a+b*x+c*x^2), q), x)];

KK3 := [coeffs(st(x^2*(a+b*x+c*x^2), q), x)];

LinearAlgebra[Transpose](Matrix([KK1, KK2,KK3]))mod 3;

end proc;

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mr34:=proc(a,b,c,d,x,q) local KK1,KK2,KK3,KK4;

KK1:=[coeffs(st(a+b*x+c*x^2+d*x^3, q), x)];

KK2 := [coeffs(st(x*(a+b*x+c*x^2+d*x^3), q), x)];

KK3 := [coeffs(st(x^2*(a+b*x+c*x^2+d*x^3), q), x)];KK4 := [coeffs(st(x^3*(a+b*x+c*x^2+d*x^3), q), x)];

LinearAlgebra[Transpose](Matrix([KK1, KK2,KK3,KK4]))mod 3;

end proc;

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References

1) David S. Dummit, Richard M. Foote , Abstract Algebra ( 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

2) Nathan Jacobson, Basic Algebra I , (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

3) Maple User Manual, Version 15.01, 2011.

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